55.3.13 problem 13

Internal problem ID [13317]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3. Equations Containing Exponential Functions
Problem number : 13
Date solved : Wednesday, October 01, 2025 at 06:59:42 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&={\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 97

ode:=diff(y(x),x) = exp(lambda*x)*y(x)^2+a*exp(x*mu)*y(x)+a*lambda*exp((mu-lambda)*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\lambda \left (a c_1 \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {2 \mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+\lambda -\mu \right )}{\left (\mu -\lambda \right ) \left (c_1 \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+{\mathrm e}^{\lambda x}\right )} \]

✓ Mathematica. Time used: 1.525 (sec). Leaf size: 166

ode=D[y[x],x]==Exp[\[Lambda]*x]*y[x]^2+a*Exp[\[Mu]*x]*y[x]+a*\[Lambda]*Exp[(\[Mu]-\[Lambda])*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {e^{\lambda (-x)} \left (-\lambda \left (-\frac {a e^{\mu x}}{\mu }\right )^{\lambda /\mu } \Gamma \left (-\frac {\lambda }{\mu },-\frac {a e^{x \mu }}{\mu }\right )+\mu e^{\frac {a e^{\mu x}}{\mu }}+c_1 \lambda e^{\frac {\lambda }{\mu }+1} \left (e^{\mu x}\right )^{\lambda /\mu }\right )}{-\left (-\frac {a e^{\mu x}}{\mu }\right )^{\lambda /\mu } \Gamma \left (-\frac {\lambda }{\mu },-\frac {a e^{x \mu }}{\mu }\right )+c_1 e^{\frac {\lambda }{\mu }+1} \left (e^{\mu x}\right )^{\lambda /\mu }}\\ y(x)&\to \lambda \left (-e^{\lambda (-x)}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
mu = symbols("mu") 
y = Function("y") 
ode = Eq(-a*lambda_*exp(x*(-lambda_ + mu)) - a*y(x)*exp(mu*x) - y(x)**2*exp(lambda_*x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*lambda_*exp(-lambda_*x + mu*x) - a*y(x)*exp(mu*x) - y(x)**2*exp(lambda_*x) + Derivative(y(x), x) cannot be solved by the factorable group method